Functional analysis: Difference between revisions

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===Hilbert spaces===

[[Hilbert space]]s can be completely classified: there is a unique Hilbert space [[up to]] [[isomorphism]] for every [[cardinal number|cardinality]] of the [[orthonormal basis]].<ref>{{Cite book| last=Riesz|first=Frigyes| url=https://www.worldcat.org/oclc/21228994|title=Functional analysis|date=1990|publisher=Dover Publications| others = Béla Szőkefalvi-Nagy, Leo F. Boron|isbn=0-486-66289-6|edition=Dover |___location=New York|oclc=21228994| pages = 195–199}}</ref> Finite-dimensional Hilbert spaces are fully understood in [[linear algebra]], and infinite-dimensional [[Separable space|separable]] Hilbert spaces are isomorphic to [[Sequence space#ℓp spaces|<math>\ell^{\,2}(\aleph_0)\,</math>]]. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to determineprove whetherthat every bounded linear operator on a Hilbert space has a proper [[invariant subspace]]. Many special cases of this [[invariant subspace problem]] have already been proven.

===Banach spaces===